Question

The boundaries of the shaded region are the x-axis, the line y = mx + c, and the curve y = f(x). Find the area of this region by writing it as a function of x and integrating with respect to x.

Answer 1

To find the area of the shaded region, set up and evaluate an integral of (f(x) - mx - c) with respect to x, with appropriate limits of integration.

Step-by-step explanation:

To find the area of the shaded region, we need to determine the limits of integration for the integral. The lower limit will be the x-coordinate of the point(s) of intersection between the curve y = f(x) and the line y = mx + c. The upper limit will be the x-coordinate of the point where the curve intersects the x-axis. Let's call these x1 and x2 respectively.

The area of the shaded region can be written as the integral of f(x) - mx - c with respect to x, integrated from x1 to x2:

A = ∫(f(x) - mx - c) dx

Using the properties of integrals, we can evaluate this integral to find the area of the shaded region.